Convergence of the Euler scheme for a class of stochastic differential equations (Q2720356)

The authors consider \(n\)-dimensional systems of autonomous Itô stochastic ordinary differential equations NEWLINE\[NEWLINE dx(t) = f(x(t)) dt + g(x(t)) dB(t), NEWLINE\]NEWLINE where \(B(t)\) is an \(m\)-dimensional Brownian motion, and corresponding Euler-Maruyama approximations. Most of the existing proofs of the convergence of the Euler-Maruyama scheme (and other methods as well) are based on the assumption that the drift and diffusion functions \(f\) and \(g\) satisfy a linear growth condition and are globally Lipschitz-continuous. These conditions are rather restrictive, as a lot of interesting nonlinear models do not meet these requirements. In this article, the authors relax these conditions, first they replace global Lipschitz-continuity by local Lipschitz-continuity. Second, instead of the linear growth condition, they give a weaker condition, using a Lyapunov-type function, ensuring that the solution will not explode in finite time. Then they prove convergence in probability of the Euler-Maruyama method. Two illustrative examples are provided, these models are studied analytically by the authors in two forthcoming articles, where also numerical case studies are performed. Thus the current paper also provides support for these case studies. A discussion of possible and interesting extensions concludes the article.